Intensity

Sound is a variation in pressure that ranges from…

P_min = P₀ − ∆P

to…

P = P₀ + ∆P

Where…

No sound wave may vary in pressure more than the static pressure of the medium itself. If it did, the minimum pressure would be less than zero, which is impossible. Sound pressure levels are computed relative to an agreed upon standard reference pressure. This number is different for gases and liquids.

1. For this part, we'll use the standard value of atmospheric pressure (101,325 Pa) and the reference pressure in air (20 μPa).
    

   LP = 20 dB log	⎛ ⎝	∆P	⎞ ⎠
   		P0
   LP = 20 dB log	⎛ ⎝	101,325 Pa	⎞ ⎠
   		20 × 10−6 Pa
   LP = 194 db

2. For this part, we need to first compute the pressure at depth.
    

   P = P0 + ρgh

   P = (101,325 Pa) + (1,025 kg/m3)(9.8 m/s2)(1,000 m)

   P = 10,146,325 Pa

   Then we need to compare this pressure to the reference pressure in water (1 µPa).

   LP = 20 dB log	⎛ ⎝	∆P	⎞ ⎠
   		P0
   LP = 20 dB log	⎛ ⎝	10,146,325 Pa	⎞ ⎠
   		1 × 10−6 Pa
   LP = 260 db

The solutions.

1. For pressure amplitude, compare 60 dB to the reference intensity of 20 μPa at 0 dB. Start with the equation for pressure level in decibels.
    

   I =	∆P2
   	2ρv

   Solve for ∆P.

   LP 20 dB	= log  ∆P ∆P0
   10LP/20 dB	=  ∆P ∆P0
   ∆P	= ∆P010L/20 dB

   Substitute and compute (a.k.a. plug and chug).

   ∆P = (20 µPa)1060 dB/20 dB

   ∆P = (20 µPa)103

   ∆P = 20 mPa

   Just a three place decimal shift written as a prefix change from micro to milli.

2. We have an equation that relates density amplitude to pressure amplitude.
    

   ∆ρ =	∆P
   	v2

   Let's assume the conversation is taking place in air at room temperature (20 ℃) with a speed of sound of 343 m/s. Divide and conquor!

   ∆ρ =	(0.020 Pa)
   	(343 m/s)2
   ∆ρ = 2.5 × 10−7 kg/m3

   I can't relate to numbers this small.

3. Before we can compute the displacement amplitude, we'll need the intensity. The density of air at room temperature (20 ℃) is 1.207 kg/m³.
    

   I =	∆P2 2ρv
   I =	(0.020 Pa)2
   	2(343 m/s)(1.207 kg/m3)
   I =	4.8 × 10−7 W/m2

   Intensity allows us to find displacement amplitude.

   I =	2π2ρƒ2vΔx2
   Δx =	√	I
   		2π2ρƒ2v

   Human speech doesn't occur at just one frequency. Early Twentieth Century telephone engineers determined that most of the power in human speech fit between 300 and 3,400 Hz. Let's take the geometric mean of this range…

   √(300 Hz × 3400 Hz) ≈ 1000 Hz

   …and continue.

   Δx =	√	4.8 × 10−7 W/m2
   		2π2(1.207 kg/m3)(1,000 Hz)2(343 m/s)
   Δx = 7.7 × 10−9m = 7.7 nm

   A nanometer is about ten atoms in a row. This means the air molecules driving your eardrum only wiggle back and forth a distance of about 150 atoms when listening to human speech. (I doubled the displacement amplitude since they are moving that distance on both side of their equilibrium position.)

4. Velocity amplitude is most easily computed from displacement amplitude.
    

   ∆v = 2πƒ ∆x

   ∆v = 2π(1000 Hz)(7.7 × 10−9m)

   ∆v = 4.8 × 10−5 m/s

   Another small number that means nothing to me.

5. For acceleration amplitude.
    

   ∆a = 2πƒ ∆v

   ∆a = 2π(1000 Hz)(4.8 × 10−5m/s)

   ∆a = 0.30 m/s2

   Not such a small number anymore. Not a lot when compared to gravity, but not microscopic like the other amplitudes.

Answer it.

Answer it.